Optimal. Leaf size=591 \[ -\frac {45 b^3 n^3 x^2}{128 d f}+\frac {3}{64} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{128 d^2 f^2}-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.50, antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2504, 2442,
45, 2424, 2342, 2341, 2421, 2430, 6724, 2423, 2438} \begin {gather*} \frac {3 b^2 n^2 \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{16 d^2 f^2}+\frac {3 b^2 n^2 \text {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}-\frac {3 b n \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {PolyLog}\left (2,-d f x^2\right )}{64 d^2 f^2}-\frac {3 b^3 n^3 \text {PolyLog}\left (3,-d f x^2\right )}{32 d^2 f^2}-\frac {3 b^3 n^3 \text {PolyLog}\left (4,-d f x^2\right )}{16 d^2 f^2}-\frac {3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{32 d^2 f^2}+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}+\frac {3}{32} b^2 n^2 x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{16 d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{16} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b^3 n^3 \log \left (d f x^2+1\right )}{128 d^2 f^2}-\frac {45 b^3 n^3 x^2}{128 d f}-\frac {3}{128} b^3 n^3 x^4 \log \left (d f x^2+1\right )+\frac {3}{64} b^3 n^3 x^4 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2341
Rule 2342
Rule 2421
Rule 2423
Rule 2424
Rule 2430
Rule 2438
Rule 2442
Rule 2504
Rule 6724
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(3 b n) \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )\right ) \, dx\\ &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac {1}{8} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {1}{4} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right ) \, dx+\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x} \, dx}{4 d^2 f^2}-\frac {(3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 d f}\\ &=-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {1}{16} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{2} \left (3 b^2 n^2\right ) \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )\right ) \, dx+\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{x} \, dx}{4 d^2 f^2}+\frac {\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 d f}\\ &=-\frac {3 b^3 n^3 x^2}{16 d f}+\frac {3}{256} b^3 n^3 x^4+\frac {3 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}-\frac {3}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {1}{16} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{8} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right ) \, dx-\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x} \, dx}{8 d^2 f^2}+\frac {\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 d f}-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-d f x^2\right )}{x} \, dx}{8 d^2 f^2}\\ &=-\frac {9 b^3 n^3 x^2}{32 d f}+\frac {3}{128} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac {1}{8} \left (3 b^3 n^3\right ) \int \left (\frac {x}{4 d f}-\frac {x^3}{8}-\frac {\log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \log \left (1+d f x^2\right )\right ) \, dx-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-d f x^2\right )}{x} \, dx}{16 d^2 f^2}\\ &=-\frac {21 b^3 n^3 x^2}{64 d f}+\frac {9}{256} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac {1}{32} \left (3 b^3 n^3\right ) \int x^3 \log \left (1+d f x^2\right ) \, dx+\frac {\left (3 b^3 n^3\right ) \int \frac {\log \left (1+d f x^2\right )}{x} \, dx}{32 d^2 f^2}\\ &=-\frac {21 b^3 n^3 x^2}{64 d f}+\frac {9}{256} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac {1}{64} \left (3 b^3 n^3\right ) \text {Subst}\left (\int x \log (1+d f x) \, dx,x,x^2\right )\\ &=-\frac {21 b^3 n^3 x^2}{64 d f}+\frac {9}{256} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}+\frac {1}{128} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \frac {x^2}{1+d f x} \, dx,x,x^2\right )\\ &=-\frac {21 b^3 n^3 x^2}{64 d f}+\frac {9}{256} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}+\frac {1}{128} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \left (-\frac {1}{d^2 f^2}+\frac {x}{d f}+\frac {1}{d^2 f^2 (1+d f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {45 b^3 n^3 x^2}{128 d f}+\frac {3}{64} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{128 d^2 f^2}-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.60, size = 1234, normalized size = 2.09 \begin {gather*} -\frac {-2 d f x^2 \left (32 a^3-24 a^2 b n+12 a b^2 n^2-3 b^3 n^3+48 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+96 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+96 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-24 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+32 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+d^2 f^2 x^4 \left (32 a^3-24 a^2 b n+12 a b^2 n^2-3 b^3 n^3+48 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+96 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+96 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-24 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+32 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-2 d^2 f^2 x^4 \left (32 a^3-24 a^2 b n+12 a b^2 n^2-3 b^3 n^3+12 b \left (8 a^2-4 a b n+b^2 n^2\right ) \log \left (c x^n\right )-24 b^2 (-4 a+b n) \log ^2\left (c x^n\right )+32 b^3 \log ^3\left (c x^n\right )\right ) \log \left (1+d f x^2\right )+2 \left (32 a^3-24 a^2 b n+12 a b^2 n^2-3 b^3 n^3+48 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+96 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+96 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-24 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+32 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right ) \log \left (1+d f x^2\right )+24 b n \left (8 a^2-4 a b n+b^2 n^2+4 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+16 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+8 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (\frac {1}{2} d f x^2-\frac {1}{8} d^2 f^2 x^4-d f x^2 \log (x)+\frac {1}{2} d^2 f^2 x^4 \log (x)+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )-96 b^2 n^2 \left (4 a-b n-4 b n \log (x)+4 b \log \left (c x^n\right )\right ) \left (\frac {1}{4} d f x^2 \left (1-2 \log (x)+2 \log ^2(x)\right )-\frac {1}{32} d^2 f^2 x^4 \left (1-4 \log (x)+8 \log ^2(x)\right )-\frac {1}{2} \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{2} \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-\log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-\log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )+b^3 n^3 \left (-16 d f x^2 \left (-3+6 \log (x)-6 \log ^2(x)+4 \log ^3(x)\right )+d^2 f^2 x^4 \left (-3+12 \log (x)-24 \log ^2(x)+32 \log ^3(x)\right )+64 \left (\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )\right )+64 \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right )\right )}{256 d^2 f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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